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Incorporating the Effect of Gravity Into Image‐Based Drainage Simulations on Volumetric Images of Porous Media

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Simulating drainage in volumetric images of porous materials is a key technique for studying multiphase flow and transport. Image‐based techniques based on sphere insertion are popular due to their computational efficiency and reasonable predictions, though they lack physical rigor. Since most tomograms are small, the impact of gravity on the fluid distributions has not been previously considered. With the advent of stochastically generated images of arbitrary size, and ever larger field‐of‐view images, the validity of neglecting gravity is becoming questionable. In this work, an image‐based technique that includes the effect of gravity during gravity stabilized displacements was developed and validated. Results compared favorably with analytical solutions of capillary rise in tubes, and to micromodel experiments in terms of the pseudo‐capillary pressure curves. The compactness of the invasion front was also shown to vary linearly with the inverse Bond number. Finally, a contour map of expected error as a function of image size and Bond number was generated to help identify when gravitational effects cannot be ignored. The presented algorithm utilizes only basic image processing tools and offers the same computational advantage as other image‐based sphere insertion methods.
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1. Introduction
Simulations of two-phase flow in porous media inform a wide range of applications in fields as diverse as ground-
water remediation, geologic storage of carbon dioxide, enhanced oil recovery, and electrochemistry (Hilpert
& Miller,2001; Schulz etal., 2007; Shikhov & Arns, 2015; Weishaupt etal., 2019). Continuum models are
widely used for studying two-phase flow (Amirebrahimi & Herrmann,1993; Celia etal.,1995,2015; Chaouche
etal.,1994) and are especially useful when macroscopic behavior is of interest such as for flow in deformable
porous media (Khoei etal.,2015) or electrochemical processes (Zenyuk etal.,2015). However, continuum models
offer limited insight into underlying physical phenomena occurring on the pore scale. When pore-scale informa-
tion about two-phase flow is of interest, there are several options including the Lattice-Boltzmann, volume-
of-fluid, and level-set methods, but these require a high amount of memory and computation times (Jettestuen
etal.,2013; Schulz etal.,2015). A more computationally feasible, though less rigorous alternative is to approx-
imate the fluid distributions at the pore-scale using image-analysis techniques. Morphological image opening
(MIO) and other image-based sphere insertion (IBSI) methods are well-established means of approximating fluid
distributions during quasistatic drainage in volumetric images of porous media (i.e., obtained from tomography
or serial-sectioning). Several review and research articles have demonstrated the reasonable effectiveness of this
approach by comparison to rigorous Lattice Boltzmann models (Pot etal.,2015; Vogel etal.,2005). The method
has also been shown to compare well with experimental data in terms of observed capillary pressure curves
(Sweijen etal.,2017; Thakur etal.,2019). So, although approximate, the method is quite useful and worthy of
extending to more diverse applications. IBSI (sometimes known as the full morphology method or FMM) was
first introduced by Hazlett(1995) and further developed to operate using morphological image operations by
Hilpert and Miller(2001). It has since been improved and benchmarked against other methods for simulating
flows in porous media (V. P. Schulz etal., 2015; Vogel etal., 2005). In all IBSI implementations so far, the
effects of gravity have been neglected. This omission is normally valid for many applications which are on small
enough length scales that the impact of gravity on fluid distribution is negligible. This is usually true on the
scale of most X-ray tomography images for instance. However, in applications with larger length scales, such
as sand column experiments, vertical micromodels, and large-scale electrochemical devices, gravity may play a
Abstract Simulating drainage in volumetric images of porous materials is a key technique for studying
multiphase flow and transport. Image-based techniques based on sphere insertion are popular due to their
computational efficiency and reasonable predictions, though they lack physical rigor. Since most tomograms
are small, the impact of gravity on the fluid distributions has not been previously considered. With the advent
of stochastically generated images of arbitrary size, and ever larger field-of-view images, the validity of
neglecting gravity is becoming questionable. In this work, an image-based technique that includes the effect of
gravity during gravity stabilized displacements was developed and validated. Results compared favorably with
analytical solutions of capillary rise in tubes, and to micromodel experiments in terms of the pseudo-capillary
pressure curves. The compactness of the invasion front was also shown to vary linearly with the inverse Bond
number. Finally, a contour map of expected error as a function of image size and Bond number was generated
to help identify when gravitational effects cannot be ignored. The presented algorithm utilizes only basic image
processing tools and offers the same computational advantage as other image-based sphere insertion methods.
CHADWICK ET AL.
© 2022. American Geophysical Union.
All Rights Reserved.
Incorporating the Effect of Gravity Into Image-Based
Drainage Simulations on Volumetric Images of Porous Media
Eric A. Chadwick1,2, Lucas H. Hammen2, Volker P. Schulz2 , Aimy Bazylak1 ,
Marios A. Ioannidis3 , and Jeff T. Gostick3
1Thermofluids for Energy and Advanced Material Laboratory, Department of Mechanical and Industrial Engineering,
Institute for Sustainable Energy, Faculty of Applied Science and Engineering, University of Toronto, Toronto, ON, Canada,
2Department of Mechanical Engineering, Electrochemistry Research Cluster, Duale Hochschule Baden-Württemberg,
Mannheim, Germany, 3Porous Materials Engineering and Analysis Lab, Department of Chemical Engineering, University of
Waterloo, Waterloo, ON, Canada
Key Points:
The effect of gravity was incorporated
in an image-based drainage algorithm
Comparison to micromodel
experiments from literature
showed excellent agreement in
pseudo-capillary pressure curves,
quantitatively predicting the slope
while qualitatively matching the
residual saturation
Expected error for neglecting gravity
was compiled into a contour plot to
estimate which images and fluid pairs
should consider gravity, with errors as
high as 50% in some cases
Correspondence to:
J. T. Gostick,
jgostick@uwaterloo.ca
Citation:
Chadwick, E. A., Hammen, L. H., Schulz,
V. P., Bazylak, A., Ioannidis, M. A.,
& Gostick, J. T. (2022). Incorporating
the effect of gravity into image-based
drainage simulations on volumetric
images of porous media. Water Resources
Research, 58, e2021WR031509. https://
doi.org/10.1029/2021WR031509
Received 29 OCT 2021
Accepted 31 JAN 2022
Author Contributions:
Conceptualization: Jeff T. Gostick
Formal analysis: Lucas H. Hammen,
Aimy Bazylak, Marios A. Ioannidis
Funding acquisition: Aimy Bazylak
Investigation: Volker P. Schulz, Marios
A. Ioannidis
Methodology: Lucas H. Hammen, Jeff
T. Gostick
Project Administration: Volker P.
Schulz, Aimy Bazylak
Resources: Volker P. Schulz, Aimy
Bazylak
Software: Eric A. Chadwick, Lucas H.
Hammen, Jeff T. Gostick
Supervision: Volker P. Schulz, Aimy
Bazylak
Validation: Eric A. Chadwick, Lucas H.
Hammen, Volker P. Schulz, Marios A.
Ioannidis, Jeff T. Gostick
Visualization: Eric A. Chadwick
Writing – original draft: Eric A.
Chadwick, Volker P. Schulz, Aimy
10.1029/2021WR031509
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